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Photogrammetry for Non-Invasive Terrestrial Position/Velocity Measurement of High-Flying Aircraft

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Photogrammetry for Non-Invasive Terrestrial Position/Velocity Measurement of High-Flying Aircraft Photogrammetry Based PV Solutions Part VI: Elevation Axis Photogrammetry for Non-Invasive Terrestrial Position/Velocity Measurement of High-Flying Aircraft Part IV: Optical Encoder & Elevation Drive James A Crawford Synopsis Part I provided a simple introduction to the big-picture objectives for this multi- phase project. Part II first looked at telescope mounts, ultimately focusing on the azimuth-elevation type mount for the project. The basic mathematics for dealing with 3-phase DC motors (e.g., Clarke and Park transformations) were introduced, along with the first ingredients for modeling and controlling the DC motors in a precision manner. The Launchpad hardware platform from Texas Instruments was selected to host the motor control algorithms. In Part III, most of the attention was focused on the mechanical design, fabrication and assembly of the telescope mount. The detailed design of the hardware changed appreciably from the first concept as better approaches were recognized during the detailed design. A first-look at low rotational speed cogging torque was also made. In this part (Part IV), attention is directed to (i) the mechanical drive details for the elevation axis utilizing finally settling on a 25:1 belt-drive step- down approach and (ii) interfacing the optical encoders [5] on the azimuth and elevation axes to an Arduino Mega2560 as an interim step to the TMS320F28379D digital signal processor. This project update has been a long time in coming primarily due to detours involving development of the elevation drive elements. Even so, the journey has been enjoyable and full of learning experiences pertaining to harmonic drives, involute gears, and associated topics. The chosen optical encoder (Avago AEAT-9000) at 17-bits has proven to be a very adjustment- delicate device to calibrate and that factor also slowed progress. For example, the associated datasheet led me to initially believe the output 17-bit value was a linear binary word, but in the end, I figured out it was a Gray-encoded value! Had there been more information on the Internet from others using this encoder, my journey would have been streamlined considerably. Copyright © 2017-2020 AM1 LLC 1 of 72 Photogrammetry Based PV Solutions Part VI: Elevation Axis 1 Project Context Most of the azimuth-elevation telescope mount was completed in the previous portion (Part III) of this project as shown in Figure 1. Several important mechanical ingredients were still not completed, however. These included: Elevation axis counter-weight assembly Azimuth axis counter-weight assembly Elevation motor concept, design, fabrication and assembly Of these mechanical items, the elevation axis motor approach has required the most development effort by far. Figure 1 Az-El telescope mount design and fabrication progress as reported in the Part III report [3] The electronic elements of the project are in their infancy at this point. Although the 3-phase driver electronics (DRV-8301) mate with the TMS320F28379D digital signal processor nicely, this cannot be said of the optical encoder electronics. As discussed later in §4.1, I decided to take a detour to first develop the encoder interface using an Arduino MEGA2560 for simplicity. Copyright © 2017-2020 AM1 LLC 2 of 72 Photogrammetry Based PV Solutions Part VI: Elevation Axis 2 Counter‐Weight Assemblies The net moment arm about elevation axis must be zero in order to demand minimum torque from the elevation drive motor. Since the telescope is offset from the elevation axis slightly as shown in Figure 2, the balancing act is a bit more involved and requires attention to two axes. L1 L2 W1 W2 L4 Elevation Axis L3 W 3 Figure 2 Moment arms about elevation axis1 Balance about the elevation axis in Figure 2 requires the net moment about the axis to always be zero. The net moment condition must also be maintained when the L1-L2 segment is tilted at an angle as shown in Figure 3. An application of engineering statics shows this condition (and consequently balance) is achieved when 11LL44 WR11cos tan WR 2 2 cos tan LW 3 3 sin 0 (1) RR12 where Wk represent point-weights with Lk and Rk being the distances shown in Figure 2. In the simple case where 0 corresponding to the L1-L2 segment being horizontal, this reduces to WL11 WL 2 2 (2) which is a necessary balancing condition. This point also illustrates the steps to be taken in balancing such an assembly: (i) with the L1-L2 arm in the horizontal position, adjust W1 or W2 to achieve (2), and then (ii) adjust weight W3 to achieve (1) for nonzero values of . In order to solve (1) for the more general case, the starting point is dictated by (2) as just stated. Equation (1) can be slightly re-written for the net moment error as 11LL44 MWe ,311 WR cos tan WR 22 cos tan LW 33 sin (3) RR12 1 From U26802 Diagrams.vsd. Copyright © 2017 – 2020 AM1 LLC 3 of 72 Photogrammetry Based PV Solutions Part VI: Elevation Axis L1 L2 W1 W2 L4 R1 R2 L3 W 3 Figure 3 Repeat of Figure 2 with additional values shown This error function is shown in the case of W1 = 2, L1 = 1, L2 = 2, and L3 = 0.50 for several values of W3 in Figure 4. Since all of the curves have a positive slope, the exact solution for W3 can be found by setting the derivative of (3) to zero thereby producing WR11sinab WR 2 2 sin W3 (4) L3 1 L4 1 L4 where a tan and b tan . R1 R2 Moment Error Vs Counter-Weight and Angle 2 W =5 3 1.5 W =4 3 W =3 1 3 W =2 3 W =1.468 0.5 3 0 Moment Error Moment -0.5 -1 -1.5 -60 -40 -20 0 20 40 60 Angle , deg Figure 4 Net moment error example versus different counter-weight values2 2 Using u26803_counterbalacing.m. Copyright © 2017 – 2020 AM1 LLC 4 of 72 Photogrammetry Based PV Solutions Part VI: Elevation Axis 2.1 Telescope Tube Counter Weight Assembly I fabricated a counter weight holder from 1” thick 6061 aluminum as shown in Figure 5. I had decided early on not to use a large-diameter rod to support the counter weights, but instead to use two 3/8” diameter stainless steel rods as shown in the figure. This approach led to a better counter weight torque to physical weight ratio. Figure 5 Counter weight holder with weights. Partially installed code wheel and reader shown on the left- hand side of the elevation axis. 2.2 Counter Weights I needed a more dense material than aluminum for the elevation counter-weight. I also wanted the selected material to be durable. I happened onto several lengths of 2.75” diameter 15-5 stainless steel material which I was told were remnants for Boeing 727 landing gear axles. I turned the material on my manual lathe primarily just to clean up the lateral surface somewhat. This also entailed drilling a center hold in each cylinder of material first. Drilling the two additional holes for the 3/8” stainless steel rod holders was a little more challenging than first anticipated. The slightest amount of vibration led to substantial drill-bit walking (and dulling) on the surface even though I had pre-drilled locations with a center-stub drill. I ultimately opted to use one of the chucks from my lath to firmly secure the 15-5 stock material with the entire assembly secured in a Kurt vice as shown in Figure 6. Even the slightest amount of vibration proved to be problematic and this point cannot be over emphasized. Copyright © 2017 – 2020 AM1 LLC 5 of 72 Photogrammetry Based PV Solutions Part VI: Elevation Axis Figure 6 15-5 stock material secured in lathe chuck Figure 7 Telescope mount with counterweights further secured in a Kurt vice temporarily installed Figure 8 One counter weight with finished drilling (3/8” holes). The small piece of rectangular aluminum bar stock underneath the 15-5 workpiece made it possible to drill completely through the workpiece without damaging the chuck. Copyright © 2017 – 2020 AM1 LLC 6 of 72 Photogrammetry Based PV Solutions Part VI: Elevation Axis 3 Elevation Drive Unlike the azimuth drive for the project, the elevation drive needs to use a much lower torque motor to keep its size down. As such, the motor torque needs to be increased by some mechanical means. Generally speaking, motor torque is proportional to motor RPM, but the RPM is essentially zero for a positioning-only application like this. 3.1 Introduction to Harmonic Drives The motivation for looking at the harmonic drive concept in the first place is that it offers (i) potentially zero back-lash, (ii) high reduction ratios, (iii) good torque efficiency, and (iv) potentially small size. Commercial harmonic drives offer excellent performance but are also quite expensive ranging from $280 to more commonly $1,000 or more each. Purchasing ready-made harmonic drives would more or less defeated the purpose of having my own precision mill too. A break-out picture of a commercially available harmonic drive is shown in Figure 9. 3.2 Baseline Trial The most difficult component to obtain for a harmonic drive is unquestionably the flexspline. The flexspline shown in Figure 9 does not lend itself to easy machining. Rather than attempt to use a fully geared design as shown in this figure, I initially opted to pursue a friction-based design and subsequently found someone else who had done this earlier and patented it [10]. A conceptual approach from this patent is shown in Figure 10.
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